Higher Auslander Algebras arising from Dynkin Quivers and n-Representation Finite Algebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914197638479872 |
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| author | Sen, Emre |
| author_facet | Sen, Emre |
| contents | In the derived category of mod-KQ for Dynkin quiver Q, we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $3k+2$ for any $k\geq 1$. Furthermore, we extend this construction for higher analogues of representation finite and hereditary algebras. Specifically, if $M$ is an n-cluster tilting object in the bounded derived category of n-representation finite and n-hereditary algebra, then we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $(n+2)k+n+1$ for any $k\geq 1$.
As an application, we revisit the higher Auslander correspondence. Firstly, we describe the corresponding module categories that have higher cluster-tilting objects, and then we discuss their relationship with certain full subcategories of the derived category. Consequently, we obtain a vast list of n-representation finite n-hereditary algebras whose n-cluster tilting objects are always minimal generator-cogenerator. Moreover, resulting algebras can be realized as endomorphism algebras of certain full subcategories of (higher) cluster categories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_13262 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher Auslander Algebras arising from Dynkin Quivers and n-Representation Finite Algebras Sen, Emre Representation Theory Category Theory Rings and Algebras 16E05, 16E10, 16E35, 16G20, 16G70 In the derived category of mod-KQ for Dynkin quiver Q, we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $3k+2$ for any $k\geq 1$. Furthermore, we extend this construction for higher analogues of representation finite and hereditary algebras. Specifically, if $M$ is an n-cluster tilting object in the bounded derived category of n-representation finite and n-hereditary algebra, then we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $(n+2)k+n+1$ for any $k\geq 1$. As an application, we revisit the higher Auslander correspondence. Firstly, we describe the corresponding module categories that have higher cluster-tilting objects, and then we discuss their relationship with certain full subcategories of the derived category. Consequently, we obtain a vast list of n-representation finite n-hereditary algebras whose n-cluster tilting objects are always minimal generator-cogenerator. Moreover, resulting algebras can be realized as endomorphism algebras of certain full subcategories of (higher) cluster categories. |
| title | Higher Auslander Algebras arising from Dynkin Quivers and n-Representation Finite Algebras |
| topic | Representation Theory Category Theory Rings and Algebras 16E05, 16E10, 16E35, 16G20, 16G70 |
| url | https://arxiv.org/abs/2307.13262 |